Complex Numbers

Complex numbers are helpful in finding the square root of negative numbers. The concept of complex numbers was first referred to in the 1st century by a Greek mathematician, Hero of Alexandria when he tried to find the square root of a negative number. But he merely changed the negative into positive and simply took the numeric root value. Further, the real identity of a complex number was defined in the 16th century by Italian mathematician Gerolamo Cardano, in the process of finding the negative roots of cubic and quadratic polynomial expressions.

Definition

A complex number is the sum of a real number and an imaginary number. A complex number is of the form $a+ib$ and is usually represented by $z$. $z=a+ib$

Here both $a$ , $b$ are real numbers and $i={\sqrt {-1}}$. The value '$a$' is called the real part which is denoted by $Re\ z$, and '$b$' is called the imaginary part $Im\ z$.  Also, $ib$ is called an imaginary number.

Examples of Complex Numbers:

$z=2+i3$

$z=-1+i{\sqrt {3}}$

$z=4+i\left[{\frac {-1}{11}}\right]$

Representation of a Complex Number

Fig. 1 - Complex Numbers

The way Complex Number is represented is shown in the fig.1.

For example: $z=2+i5$ , then

Real Part: $Re\ z=2$

Imaginary Part: $Im\ z=5$

Equality of complex numbers

Two complex numbers $z_{1}=a+ib$ and $z_{2}=c+id$ are equal if $a=c$ and $b=d$

Example: if $5x+i(4x-y)=4+i(-5)$ where x and y are real numbers, then find the values of x and y.

$5x+i(4x-y)=4+i(-5)................(1)$

Equating the real and the imaginary parts of $(1)$ we get

$5x=4$ Hence $x={\frac {4}{5}}$

$4x-y=-5$

$4\left[{\frac {4}{5}}\right]-y=-5$

$y=4\left[{\frac {4}{5}}\right]+5$

$y={\frac {16+25}{5}}={\frac {41}{5}}$

Answer: $x={\frac {4}{5}}$ and $y={\frac {41}{5}}$